B. Weaver (31-Oct-2005) Probability & Hypothesis Testing 1 Probability and Hypothesis Testing 1.1 PROBABILITY AND INFERENCE The area of descriptive statistics is concerned with meaningful and efficient ways of presenting data. When it comes to inferential statistics, though, our goal is to make some statement about a characteristic of a population based on what we know about a sample drawn from that population. Generally speaking, there are two kinds of statements one can make. One type concerns parameter estimation, and the other hypothesis testing. Parameter Estimation In parameter estimation, one is interested in determining the magnitude of some population characteristic. Consider, for example an economist who wishes to estimate the average monthly amount of money spent on food by unmarried college students. Rather than testing all college students, he/she can test a sample of college students, and then apply the techniques of inferential statistics to estimate the population parameter. The conclusion of such a study would be something like: The probability is 0.95 that the population mean falls within the interval of £130-£150. Hypothesis Testing In the hypothesis testing situation, an experimenter wishes to test the hypothesis that some treatment has the effect of changing a population parameter. For example, an educational psychologist believes that a new method of teaching mathematics is superior to the usual way of teaching. The hypothesis to be tested is that all students will perform better (i.e., receive higher grades) if the new method is employed. Again, the experimenter does not test everyone in the population. Rather, he/she draws a sample from the population. Half of the subjects are taught with the Old method, and half with the New method. Finally, the experimenter compares the mean test results of the two groups. It is not enough, however, to simply state that the mean is higher for New than Old (assuming that to be the case). After carrying out the appropriate inference test, the experimenter would hope to conclude with a statement like: The probability that the New-Old mean difference is due to chance (rather than to the different teaching methods) is less than 0.01. Note that in both parameter estimation and hypothesis testing, the conclusions that are drawn have to do with probabilities. Therefore, in order to really understand parameter estimation and hypothesis testing, one has to know a little bit about basic probability. 1.2 RANDOM SAMPLING Random sampling is important because it allows us to apply the laws of probability to sample data, and to draw inferences about the corresponding populations.

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B. Weaver (31-Oct-2005) Probability & Hypothesis Testing 2 Sampling With Replacement A sample is random if each member of the population is equally likely to be selected each time a selection is made. When N is small, the distinction between with and without replacement is very important. If one samples with replacement, the probability of a particular element being selected is constant from trial to trial (e.g., 1/10 if

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